Mathematics and Statistics for AI

Represent learners as feature vectors and check compatible matrix shapes.

Evidence: Correct dimensions for three matrix operations with explanations.

Implement dot products and transformations on a small dataset.

Evidence: Manual and notebook results agree within floating-point tolerance.

Plot linear and nonlinear functions; approximate slopes numerically.

Evidence: A slope plot and explanation of the effect of step size.

Calculate conditional probabilities from a contingency table.

Evidence: All supplied probability checks pass and denominators are identified.

Simulate Bernoulli and normal data and compare empirical summaries.

Evidence: Plots and estimates distinguish sample variation from model assumptions.

Update an event probability from a diagnostic observation.

Evidence: A worked posterior with explicit prior and likelihood assumptions.

Compare random and convenience samples from a known synthetic population.

Evidence: A report showing the direction of bias and a better sampling plan.

Bootstrap a mean and interpret uncertainty without treating it as certainty.

Evidence: A reproducible interval and a correct interpretation of its procedure.

Compare two groups; separate effect size from statistical significance.

Evidence: A test report with assumptions, effect size and practical relevance.

Compute squared error and cross entropy for supplied predictions.

Evidence: Correct loss calculations and a reasoned choice for each task.

Implement updates for a one-feature regression problem.

Evidence: A decreasing loss trace and a diagnosis of an excessive learning rate.

Audit a misleading AI performance claim using sample size and uncertainty.

Evidence: A corrected conclusion with at least three justified caveats.